thermodynamics-statistical-mechanics

name	thermodynamics_statistical_mechanics
description	Use when computing partition functions, thermodynamic potentials, phase transitions, equations of state, or ensemble averages.

Thermodynamics and Statistical Mechanics

Apply this skill when the problem involves thermal equilibrium, partition functions, free energies, entropy, equations of state, phase transitions, or the statistical behavior of many-particle systems.

Goal

Compute thermodynamic quantities from microscopic models using ensemble theory, or apply thermodynamic identities and potentials to macroscopic systems.

Scope

• Microcanonical, canonical, and grand canonical ensembles
• Partition functions and free energies (Helmholtz F, Gibbs G, grand potential Omega)
• Equations of state and thermodynamic response functions (heat capacity, compressibility, susceptibility)
• Classical ideal gas, quantum ideal gases (Bose-Einstein, Fermi-Dirac)
• Phase transitions: Ehrenfest classification, Landau theory, critical exponents, mean-field theory
• Fluctuations and fluctuation-dissipation theorem
• Entropy: Boltzmann, Gibbs, Shannon, von Neumann
• Monte Carlo methods: Metropolis algorithm, importance sampling

Inputs

• microscopic_model: Hamiltonian or energy function of the system
• ensemble: Which ensemble to use (microcanonical, canonical, grand canonical)
• control_parameters: Temperature T, pressure P, chemical potential mu, external fields
• particle_statistics: Classical, bosonic, or fermionic

Outputs

• partition_function: Z (or its logarithm) in the appropriate ensemble
• thermodynamic_potentials: F, G, Omega, S, U as functions of control parameters
• equations_of_state: Pressure, density, magnetization as functions of T, V, N, etc.
• phase_diagram: Location of phase boundaries and critical points (if applicable)

Workflow

1. Identify the ensemble.

   • Fixed E, V, N -> microcanonical (Omega = k_B ln W).
   • Fixed T, V, N -> canonical (Z = sum exp(-beta E_i), F = -k_B T ln Z).
   • Fixed T, V, mu -> grand canonical (Xi = sum exp(-beta(E_i - mu N_i))).

2. Compute the partition function.

   • Sum (or integrate) over all microstates.
   • For quantum systems, include the correct statistics (Bose-Einstein or Fermi-Dirac).
   • Use saddle-point / steepest descent for large N when exact evaluation is intractable.

3. Derive thermodynamic quantities.

   • Internal energy: U = -partial(ln Z)/partial(beta).
   • Entropy: S = -partial F / partial T.
   • Pressure: P = -partial F / partial V.
   • Heat capacity: C_V = partial U / partial T at constant V.
   • Use Maxwell relations to connect different response functions.

4. Analyze phase transitions (if applicable).

   • Look for non-analyticities in the free energy as a function of control parameters.
   • Landau theory: expand free energy in an order parameter near the critical point.
   • Compute critical exponents (mean-field or beyond).

5. Numerical methods (if needed).

   • Monte Carlo simulation with Metropolis algorithm for interacting systems.
   • Molecular dynamics for time-dependent thermodynamic properties.

Quality Checks

• The free energy must be extensive (proportional to system size) in the thermodynamic limit.
• Entropy must be non-negative and satisfy the third law (S -> 0 as T -> 0 for non-degenerate ground states).
• Heat capacity must be non-negative: C_V >= 0.
• Thermodynamic identities (Maxwell relations) must be self-consistent.
• In the high-temperature limit, quantum results should reduce to classical results.

Constraints

• Do not use the canonical ensemble for systems that exchange particles with a reservoir; use the grand canonical ensemble.
• Mean-field theory gives incorrect critical exponents near phase transitions in low dimensions (d <= 4 for Ising-like systems); state this limitation explicitly.
• For quantum gases, do not neglect quantum statistics (Bose-Einstein condensation, Fermi surface) at temperatures comparable to the degeneracy temperature.
• Always state whether the calculation is in the thermodynamic limit (N -> infinity, V -> infinity, N/V fixed) or for a finite system.